A logarithm, in mathematics, is the exponent or power to which a number needs to be raised for producing that number. For example, the logarithm of 10,000 to the base 10 is 4, as 10 needs to be raised to the power of 4 to get the number 10,000 (104=10,000). Real and positive numbers have real number logarithms, whereas complex and negative numbers have complex logarithms. A logarithmic function is defined as follows:
If b is a number such that b > 0, 
and x > 0
then, 
is equivalent to 
This is usually read as “log base b of x“.
Logarithms are used to help perform complicated calculations and are applied in various fields such as chemistry, physics, statistics, computer science, economics, astronomy, engineering and music. Evaluating logarithms is very common today as it plays a crucial role in solving problems related to variable quantities and exponents.
Before we learn how to evaluate logarithms, we must understand the essential rules to be followed while evaluating them without using a calculator. Without these following rules, it will not be possible for you to evaluate logarithms.
1. Power Rule
log a x n = n log a x
2. Product Rule
log a x y = log a x + log a y
3. Change of base Rule
Log a b = log c b / log c a
4. Quotient Rule
log a x / y = log a x – log a y
5. Log 0 to any base always equals 1.
6. If a = b y, then y = log b (a)
7. Log 1 to any base is always equal to 0.
8. When the value and the base are same, for example log1010, it will always be equal to 1.
The most commonly used base for solving logarithms is 10 and if there is no base mentioned then you must always take the base to be 10.
Here are some examples of evaluating logarithms without using calculator.
- Evaluate 2 log3 27 + log3 81 – 3 log3 9
Answer:
2 log3 27 + log3 81 – 3 log3 9
= 2 log3 33 + log3 34 – 3 log3 32
= 3*2log3 3 + 4log3 3 – 2*3log3 3
Now, by using the power rule we get
= 6log3 3 +4log3 3– 6log3 3
=6(1) + 4(1) -6(1)
=10-6
=4.
- Evaluate 2 log3 5 + 2log3 5 – log3 625
Answer:
log6 36 + log3 243 – 2 log5 25
= log6 62 + log3 35 – log5 52
Now, by using the power rule we get
= 2log6 6 + 5log3 3 – 2log5 5
= 2(1) + 5(1) – 2(1)
= 2+5-2
= 7-2 =5
You can also use an online calculator to solve any kind of logarithms. It is very simple and easy to use. The websites providing these online calculators also give exercise questions and study guides for you to practice solving them.
You will need to follow these steps to evaluate logarithms online:
- First, go to any website that has an online calculator. You can visit any of these following websites:
http://www.1728.com/logrithm.htm
http://rechneronline.de/logarithm/
http://www.math.utah.edu/~pa/math/Log.html
- Open the website and click on the logarithm button. Enter the “logarithm of” value in the first box. In the box right next to it, type the number for which you need to find the algorithm.
- Now, choose the base using the radio buttons. If you don’t select a base, 10 will be selected as the default base. If the base you want is not listed in the section you can click on the box labeled “other” and fill in you base number.
- Now complete the formula and click on “enter” to get your answer, which appears in the final box.
- You can also calculate antilogarithm by clicking on the “Anti-log” button.
- Among all these websites, you can bookmark the one which you like the most for easy access next time.
Tips:
- You need to check with you teachers if you are allowed to use calculators for your homework assignments.
- Remember that calculators are only for convenience; you must learn all the principles thoroughly and practice solving them by hand.
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